Which problems have strongly exponential complexity?

Exact Algorithms for NP-Hard Problems: A Survey

“…Since the standard reductions between NP-complete problems may increase the instance sizes, many questions in computational complexity theory depend delicately on the choice of parameters. The right approach seems to be to include an explicit complexity parameter in the problem specification (Impagliazzo, Paturi & Zane [21]). Recall that the decision version of every problem in NP can be formulated in the following way:…”

“…There are two natural complexity parameters for k-satisfiability, the number of logical variables and the number of clauses. Impagliazzo, Paturi & Zane [21] prove that the two variants of k-satisfiability with these two complexity parameters are SERF-reducible to each other, and hence are equivalent under SERF-reductions. This indicates that for k-satisfiability the exact parameterization is not very important, and that all natural parameterizations of k-satisfiability should be SERF-reducible to each other.…”

“…This indicates that for k-satisfiability the exact parameterization is not very important, and that all natural parameterizations of k-satisfiability should be SERF-reducible to each other. Most important, the paper [21] shows that for any fixed k ≥ 3 the k-satisfiability problem is SNPcomplete under SERF-reductions. As we discussed above, this implies that for any fixed k ≥ 3 the k-satisfiability problem cannot have a sub-exponential time algorithm, unless SNP ⊆ SUBEXP.…”

“…Now let us discuss the behavior of some other problems in NP. Impagliazzo, Paturi & Zane [21] show that for any fixed k ≥ 3 the k-colorability problem is SNP-complete under SERF-reductions. Hence 3-colorability can not be solved in sub-exponential time, unless SNP ⊆ SUBEXP.…”

On the complexity of finding balanced oneway cuts

“…The reduction is composed of the famous reduction from 3SAT to clique [7] (associate a vertex to each literal occurrence in each clause and draw edges between vertices from different clauses unless they correspond to opposite literals), which results in a graph with O(m) vertices, followed by the previously discussed reductions from clique to balanced biclique and therefrom to oneway bisection, both of which are quadratic in size. We note here 3SAT has algorithms that are subexponential in the number of clauses m only if it has algorithms that are subexponential in the number of variables n. See [5]. 2 Theorem 3 Let G = (V, E) be a directed graph with a b-balanced oneway cut (S, T ) for some b < 1 2 .…”